Finite Group Databasebeta

← #11233586

Subgroups — #11233586

Order 768.

1638 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2402C211yesmaximal, characteristic, complemented
2#78369C214yesmaximal, complemented
3#78277C214yesmaximal, complemented
4#77790C214yesmaximal, complemented
5#77772C214yesmaximal, complemented
6#73800C211yesmaximal, characteristic
7#73721C211yesmaximal, characteristic, complemented
8#73717C212yesmaximal, complemented
9#73692C212yesmaximal
10#73673C212yesmaximal
11#73645C211yesmaximal, characteristic
12#73640C212yesmaximal, complemented
13#73637C211yesmaximal, characteristic, complemented
14#73391C211yesmaximal, characteristic, complemented
15#73387C211yesmaximal, characteristic
16#45350—31nomaximal, Sylow
17C2×#898C2214yes—
18C2×#897C2214yes—
19C2×#890C2214yes—
20C2×#875C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6C2212yes—
24C2×C4×Dic6C2211yescharacteristic
25C3×#400C2212yes—
26C3×#400C2212yes—
27C3×#398C2212yes—
28C3×#398C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C4⋊C4C2212yes—
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×C4×C4⋊C4C2212yes—
33C3×#376C2211yescharacteristic
34#4994C2214yes—
35#4993C2214yes—
36#4990C2214yes—
37#4990C2214yes—
38#4984C2214yes—
39#4984C2214yes—
40#4977C2214yes—
41#4976C2214yes—
42Dic3×C4⋊C4C2214yes—
43#4939C2212yes—
44#4939C2211yescharacteristic
45#4939C2212yes—
46#4936C2211yescharacteristic
47#4935C2212yes—
48#4935C2211yescharacteristic
49#4935C2212yes—
50C4×C4⋊Dic3C2212yes—
51C4×C4⋊Dic3C2212yes—
52#4929C2211yescharacteristic
53#4929C2212yes—
54#4928C2212yes—
55#4928C2212yes—
56#4677C2212yes—
57#4677C2212yes—
58#4675C2214yes—
59#4674C2214yes—
60#4670C2214yes—
61#4670C2214yes—
62#4669C2214yes—
63#4668C2214yes—
64#4666C2212yes—
65#4664C2212yes—
66#4661C2214yes—
67#4661C2214yes—
68#4656C2212yes—
69#4656C2212yes—
70#4655C2214yes—
71#4655C2214yes—
72#4654C2214yes—
73#4650C2212yes—
74#4649C2212yes—
75#4646C2214yes—
76#2773—34no—
77#2698—34no—
78#2567—31no—
79#2556—34no—
80#2552—32no—
81#2542—31no—
82#2519—34no—
83#2512—32no—
84#2448—32no—
85#2402D311yescharacteristic, complemented
86#2357—31no—
87#2355—32no—
88#2351—31no—
89#2331—31no—
90#2326—31no—
91C22×D12C2311yescharacteristic
92C22×C4×D3C2312yes—
93C22×Dic6C2311yescharacteristic
94C22×Dic6C2312yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2312yes—
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2312yes—
105C2×C3×C42C2311yescharacteristic
106C2×(C2×D3)⋊C4C2314yes—
107C2×(C2×D3)⋊C4C2314yes—
108C2×(C2×D3)⋊C4C2314yes—
109C2×(C2×D3)⋊C4C2312yes—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×C4⋊Dic3C2312yes—
114C2×C4⋊Dic3C2314yes—
115C2×C4⋊Dic3C2312yes—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2312yes—
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2314yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2314yes—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2314yes—
125C2×Dic3⋊C4C2312yes—
126C2×Dic3⋊C4C2314yes—
127C2×C4×Dic3C2312yes—
128C2×C4×Dic3C2314yes—
129C2×C4×Dic3C2314yes—
130C2×C4×Dic3C2314yes—
131#898—48no—
132#897—48no—
133#890—48no—
134#875—24no—
135C4×D12—24no—
136#869—24no—
137C4×Dic6—28no—
138C4×Dic6—24no—
139C3×#95C2312yes—
140C3×#95C2311yescharacteristic
141C3×#95C2311yescharacteristic
142C3×#95C2312yes—
143C3×#95C2312yes—
144C3×#95C2312yes—
145C3×#95C2311yescharacteristic
146C3×#95C2311yescharacteristic
147C3×#95C2312yes—
148C3×#95C2312yes—
149C3⋊#95C2312yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2312yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C2×C4⋊Q8—31no—
159C2×#126—34no—
160C2×#125—34no—
161C2×#124—31no—
162C2×C22⋊Q8—34no—
163C2×C4×Q8—31no—
164C2×C4×Q8—32no—
165C2×C4×D4—31no—
166#400C2×D312yes—
167#400C2×D312yes—
168#399—34no—
169#399—32no—
170#399—32no—
171#398—34no—
172#398—34no—
173#398C2×D312yes—
174#398C2×D312yes—
175#397—34no—
176#397—34no—
177#396—34no—
178#395—32no—
179#395—32no—
180#391—32no—
181#391—31no—
182#391—34no—
183#390—31no—
184#390—32no—
185#390—32no—
186#389—34no—
187#389—34no—
188#389—34no—
189#389—32no—
190#389—32no—
191#388—34no—
192#388—34no—
193#388—34no—
194#388—34no—
195#388—34no—
196#387—34no—
197#387—34no—
198#387—34no—
199#385—34no—
200#384—32no—
201#384—31no—
202#384—32no—
203#382—34no—
204#382—31no—
205#382—32no—
206#381—32no—
207C4×C4⋊C4—34no—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—32no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4C2×D311yescharacteristic
212C4×C4⋊C4C2×D311yescharacteristic
213C4×C4⋊C4C2×D312yes—
214C4×C4⋊C4C2×D312yes—
215C4×C22⋊C4—32no—
216#376C2×D311yescharacteristic
217C23×D3C2412yes—
218C22×C3×C4C2412yes—
219C22×C3×C4C2412yes—
220C22×C3×C4C2412yes—
221C22×C3×C4C2412yes—
222C22×C3×C4C2411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×C3×C4C2412yes—
225C22×C3×C4C2412yes—
226C22×C3×C4C2411yescharacteristic
227C22×Dic3C2414yes—
228C22×Dic3C2412yes—
229C22×Dic3C2414yes—
230C22×Dic3C2412yes—
231C22×Dic3C2412yes—
232C2×D12—22no—
233C2×D12—22no—
234C2×D12—22no—
235C2×C4×D3—24no—
236C2×C4×D3—44no—
237C2×Dic6—22no—
238C2×Dic6—22no—
239C2×Dic6—24no—
240C2×Dic6—44no—
241C2×Dic6#4114yes—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C42—24no—
249C3×C42—22no—
250C3×C42—24no—
251C3×C42—22no—
252C3×C42#4114yes—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—28no—
255(C2×D3)⋊C4—28no—
256(C2×D3)⋊C4—28no—
257(C2×D3)⋊C4—44no—
258(C2×D3)⋊C4—44no—
259(C2×D3)⋊C4—44no—
260C4⋊Dic3—22no—
261C4⋊Dic3—24no—
262C4⋊Dic3—24no—
263C4⋊Dic3—24no—
264C4⋊Dic3—22no—
265C4⋊Dic3—28no—
266C4⋊Dic3—44no—
267C4⋊Dic3—44no—
268C4⋊Dic3—44no—
269C4⋊Dic3—44no—
270Dic3⋊C4—24no—
271Dic3⋊C4—28no—
272Dic3⋊C4—28no—
273Dic3⋊C4—44no—
274C4×Dic3—24no—
275C4×Dic3—28no—
276C4×Dic3—44no—
277C4×Dic3—44no—
278C22×Q8—31no—
279C22×Q8—32no—
280C22×D4—31no—
281C23×C4—32no—
282C4⋊Q8—64no—
283#126—128no—
284#125—128no—
285#124—64no—
286C22⋊Q8—128no—
287C4×Q8—68no—
288C4×Q8—64no—
289C4×D4—64no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4—34no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—34no—
296C2×C4⋊C4—34no—
297C2×C4⋊C4—32no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—34no—
300C2×C4⋊C4—34no—
301C2×C4⋊C4—34no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4C22×D312yes—
305C2×C4⋊C4C22×D312yes—
306C2×C4⋊C4C22×D312yes—
307C2×C4⋊C4C22×D312yes—
308C2×C4⋊C4C22×D312yes—
309C2×C4⋊C4C22×D312yes—
310C2×C22⋊C4—32no—
311C2×C22⋊C4—34no—
312C2×C22⋊C4—34no—
313C2×C22⋊C4—34no—
314C2×C22⋊C4—34no—
315C2×C22⋊C4—34no—
316C2×C22⋊C4—34no—
317C2×C42—34no—
318C2×C42—32no—
319C2×C42—34no—
320C2×C42—34no—
321C2×C42C22×D312yes—
322C2×C42C22×D311yescharacteristic
323C2×C42C22×D311yescharacteristic
324C2×C42C22×D312yes—
325C2×C42C22×D311yescharacteristic
326#95—32no—
327#95—34no—
328#95—32no—
329#95—34no—
330#95—34no—
331#95—34no—
332#95—34no—
333#95—34no—
334#95—34no—
335#95C22×D312yes—
336#95C22×D312yes—
337#95C22×D311yescharacteristic
338#95C22×D311yescharacteristic
339#95C22×D311yescharacteristic
340#95C22×D311yescharacteristic
341#95C22×D312yes—
342#95C22×D312yes—
343#95C22×D312yes—
344#95C22×D312yes—
345C23×C3C2511yescharacteristic
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—24no—
349C22×D3—24no—
350C22×D3—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4C2×#4112yes—
370C2×C3×C4C2×#4112yes—
371C2×C3×C4C2×#4112yes—
372C2×C3×C4C2×#4112yes—
373C2×C3×C4H(5,𝔽2)14yes—
374C2×C3×C42−1+414yes—
375C2×C3×C42−1+412yes—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—22no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—24no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3C2×#4114yes—
389D12—44no—
390C4×D3—84no—
391Dic6—28no—
392Dic6—84no—
393C24—32no—
394C2×Q8—34no—
395C2×Q8—62no—
396C2×Q8—64no—
397C2×Q8—62no—
398C2×Q8—124no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—62no—
402C22×C4—32no—
403C22×C4—34no—
404C22×C4—32no—
405C22×C4—34no—
406C22×C4—32no—
407C22×C4—64no—
408C22×C4—124no—
409C22×C4C23×D312yes—
410C22×C4C23×D312yes—
411C22×C4C23×D312yes—
412C22×C4C23×D312yes—
413C22×C4C23×D311yescharacteristic
414C22×C4C23×D311yescharacteristic
415C22×C4C23×D312yes—
416C22×C4C23×D312yes—
417C22×C4C23×D311yescharacteristic
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—68no—
425C4⋊C4—62no—
426C4⋊C4—64no—
427C4⋊C4—62no—
428C4⋊C4—64no—
429C4⋊C4—64no—
430C4⋊C4—64no—
431C4⋊C4—68no—
432C4⋊C4—68no—
433C4⋊C4—124no—
434C4⋊C4—124no—
435C4⋊C4—124no—
436C4⋊C4—124no—
437C4⋊C4—124no—
438C22⋊C4—68no—
439C22⋊C4—64no—
440C22⋊C4—68no—
441C22⋊C4—68no—
442C22⋊C4—124no—
443C22⋊C4—124no—
444C22⋊C4—124no—
445C42—24no—
446C42—22no—
447C42—24no—
448C42—22no—
449C42—68no—
450C42—64no—
451C42—124no—
452C42—124no—
453C42#23914yes—
454C22×C3C22×#4111yescharacteristic
455C22×C3C22×#4111yescharacteristic
456C22×C3C2×H(5,𝔽2)12yes—
457C22×C3C2×2−1+411yescharacteristic
458C22×C3C2×2−1+412yes—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—44no—
463C2×D3—44no—
464C3×C4—22no—
465C3×C4—22no—
466C3×C4—24no—
467C3×C4—24no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4#55214yes—
473Dic3—24no—
474Dic3—44no—
475Dic3—42no—
476Dic3—42no—
477Dic3—44no—
478C23—62no—
479C23—62no—
480C23—64no—
481C23—64no—
482C23—62no—
483C23C24×D311yescharacteristic
484Q8—68no—
485Q8—244no—
486D4—124no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—34no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—244no—
519C2×C4C2×#23912yes—
520C2×C4C2×#23912yes—
521C2×C4C2×#23912yes—
522C2×C4D3×#4112yes—
523C2×C4#100914yes—
524C2×C4#101014yes—
525C2×C4#101012yes—
526C2×C3C2×#55211yescharacteristic
527C2×C3#350811yescharacteristic
528C2×C3#350911yescharacteristic
529C2×C3#352212yes—
530C2×C3#355212yes—
531D3—82no—
532C22—122no—
533C22—124no—
534C22—124no—
535C22—122no—
536C22—122no—
537C22C22×#23911yescharacteristic
538C22C2×D3×#4111yescharacteristic
539C22C2×#100912yes—
540C22C2×#101011yescharacteristic
541C22C2×#101012yes—
542C4—24no—
543C4—24no—
544C4—22no—
545C4—22no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—64no—
551C4—124no—
552C4—122no—
553C4—124no—
554C4—122no—
555C4#569414yes—
556C3#4535011yescharacteristic, complemented, Sylow
557C2—242no—
558C2C2×#569411yescharacteristic
559C2#8612812yes—
560C2#8613412yes—
561C2#8626111yescharacteristic
562C2#8626211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2402C2
prime radicalC22×D12C23
hypercenter#2402D3
hypocenterC3#45350
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2402

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#2402C2semidirect
2. #78369C2semidirect
3. #78277C2semidirect
4. #77790C2semidirect
5. #77772C2semidirect
7. #73721C2semidirect
8. #73717C2semidirect
12. #73640C2semidirect
13. #73637C2semidirect
14. #73391C2semidirect
85. #2402D3semidirect
556. C3#45350semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#24022
#783692
#782772
#777902
#777722
#738002
#737212
#737172
#736922
#736732
#736452
#736402
#736372
#733912
#733872
#453503